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Erdős Problem #966

Number theory · posed by Paul Erdős, 1975 · solved

1 attempt · 1 machine check

Statement

Let k,r≥2k,r\geq 2. Does there exist a set A⊆NA\subseteq \mathbb{N} that contains no non-trivial arithmetic progression of length k+1k+1, yet in any rr-colouring of AA there must exist a monochromatic non-trivial arithmetic progression of length kk? Answered in the affirmative.

Context

Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

People

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Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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  • #1

    Attempt 1

    constructionAristotle ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    Aristotle

    Aristotle produced the construction and its proof and formalized the result; erdosproblems.com marks the problem PROVED with the proof verified in Lean.

    Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution

    Reviews

    1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      scope Lean formalization of the result

      erdosproblems.com marks the problem PROVED (LEAN): solved in the affirmative with the proof verified in Lean.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

    Discussion of this attempt

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Discussion

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